Chi-Square Critical Values: Quick Reference
Most-looked-up values at α = 0.05 (95% confidence) — right-tail critical values.
Chi-Square Value 3.841 — What It Means (df=1, α=0.05)
3.841 is the chi-square critical value at 1 degree of freedom and α = 0.05 (95% confidence). It means: if you run a chi-square test with df=1 and your test statistic exceeds 3.841, the result is statistically significant — you reject the null hypothesis. This value appears because 5% of the chi-square distribution with df=1 lies above 3.841. At 99% confidence (α=0.01), the critical value is 6.635. At 90% confidence (α=0.10), it is 2.706.
Chi-Square Critical Value: df=3, α=0.05 = 7.815
At 3 degrees of freedom and α = 0.05, the critical value is 7.815. This is commonly used in goodness-of-fit tests with 4 categories (df = 4−1 = 3) and in 2×2 contingency tables where one variable has 2 levels and another has 3 levels giving df = (2−1)(3−1) = 2 — wait, df=2 gives 5.991; for a 2×3 table, df=2. For df=3 specifically: a 4-category goodness-of-fit test, or a 2×4 contingency table (df=(2−1)(4−1)=3).
Chi-Square Critical Value Calculator
| df \ α | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 |
|---|
All values are right-tail critical values χ²α,df. Click any cell to highlight and auto-fill the calculator above.
How to Read a Chi-Square Table (Step by Step)
Reading a chi-square table (also called a chi-squared table or χ² distribution table) takes four steps:
Chi-Square Distribution Table (χ²)
The chi-square distribution table — also written as chi-squared distribution table, χ² distribution table, or chi table — is a reference giving critical values of the chi-square distribution. The distribution is a family of right-skewed curves defined entirely by degrees of freedom (df). As df increases, the distribution shifts right and becomes more symmetric.
Definition
The chi-square (χ²) distribution is always ≥ 0. Mean = df, Variance = 2×df. It is used in goodness-of-fit, independence, homogeneity, and variance tests.
Key Properties
Values are always right-tail critical values. The table shows the chi-square score where P(χ² > x) = α. Larger df = larger critical values at the same α.
Test Statistic Formula
χ² = Σ(O−E)²/E, where O = observed frequency and E = expected frequency. Always compare this calculated value against the table critical value.
One-Tailed vs Two-Tailed Chi-Square Test
One-Tailed (Right-Tail) — Most Common
Used in goodness-of-fit, independence, and homogeneity tests. Reject H₀ when χ²calc > χ²α,df.
Two-Tailed — Variance Tests
Used when testing H₀: σ² = σ₀². Split α between both tails. Use both the upper critical value (χ²α/2) and the lower (χ²1−α/2).
Chi-Square Table for Contingency Tables
In a test of independence, df = (rows − 1) × (columns − 1). Use the table to find the critical value for your specific table size.
Worked Example: 3×2 Contingency Table
Survey of 200 people: 3 age groups (rows) × 2 preference categories (columns). df = (3−1)×(2−1) = 2. At α = 0.05, critical value from chi-square table = 5.991. If χ² > 5.991, the variables are not independent.
Chi-Square Table PDF — Free Download
Download a free printable chi-square table PDF. All three versions include critical values at significance levels 0.005, 0.01, 0.025, 0.05, and 0.10 — suitable for exams, coursework, and research.
Chi-Square Critical Values: All Common Reference Points
Most frequently referenced values — right-tail at α = 0.05 and α = 0.01.
| df | α = 0.10 | α = 0.05 | α = 0.025 | α = 0.01 | α = 0.005 |
|---|---|---|---|---|---|
| 1 | 2.706 | 3.841 | 5.024 | 6.635 | 7.879 |
| 2 | 4.605 | 5.991 | 7.378 | 9.210 | 10.597 |
| 3 | 6.251 | 7.815 | 9.348 | 11.345 | 12.838 |
| 4 | 7.779 | 9.488 | 11.143 | 13.277 | 14.860 |
| 5 | 9.236 | 11.071 | 12.833 | 15.086 | 16.750 |
| 10 | 15.987 | 18.307 | 20.483 | 23.209 | 25.188 |
| 20 | 28.412 | 31.410 | 34.170 | 37.566 | 39.997 |
| 30 | 40.256 | 43.773 | 46.979 | 50.892 | 53.672 |
How to Use a Chi-Square Table for Hypothesis Testing
Goodness-of-Fit Test
Tests whether observed frequencies match expected distribution. df = k − 1. Example: testing if a six-sided die is fair: df = 6 − 1 = 5, critical value at α=0.05 is 11.071.
Test of Independence
Tests if two categorical variables are related. df = (r−1)(c−1). Example: gender vs product preference in a 2×3 table: df = 1×2 = 2, critical value = 5.991.
Variance Test
Tests whether population variance equals a specified value. df = n − 1. Uses both upper and lower critical values from the chi-square table (two-tailed).
Frequently Asked Questions About the Chi-Square Table
What is a chi-square table used for?
A chi-square table (also called a chi-squared table or χ² table) lists critical values of the chi-square distribution for different degrees of freedom and significance levels. It is used to decide whether to reject the null hypothesis in goodness-of-fit tests, tests of independence, and tests of homogeneity.
What is chi-square value 3.841 and where does it come from?
3.841 is the chi-square critical value at df=1 and α=0.05 (95% confidence level). It represents the chi-square score above which 5% of the probability falls when there is 1 degree of freedom. If your test statistic exceeds 3.841, the result is statistically significant. At α=0.01 the value is 6.635; at α=0.10 it is 2.706.
How do you find a chi-square critical value without a table?
You can use: (1) This page's interactive calculator above — enter df and α. (2) Excel: =CHISQ.INV.RT(alpha, df), e.g. =CHISQ.INV.RT(0.05,1) returns 3.841. (3) Python: scipy.stats.chi2.ppf(1-alpha, df). (4) R: qchisq(1-alpha, df). All these return the same values as the printed table.
What is the critical value at df=1, p=0.05?
At df=1 and α=0.05, the chi-square critical value is 3.841. This is the most commonly referenced chi-square value in statistics. If χ²calc > 3.841, reject H₀ at the 5% significance level.
How do you calculate degrees of freedom?
Goodness-of-fit: df = categories − 1. Test of independence: df = (rows − 1) × (columns − 1). Variance test: df = n − 1. Example: a 3×4 contingency table has df = (3−1) × (4−1) = 2 × 3 = 6.
Can a chi-square value be negative?
No. Chi-square values are always ≥ 0 because the formula χ² = Σ(O−E)²/E involves squared differences. The minimum value 0 occurs when observed frequencies perfectly match expected frequencies.
What does it mean when my statistic exceeds the critical value?
Reject the null hypothesis. The result is statistically significant — observed data differs from expected by more than chance at your chosen α level.